# Combinatorics

## What is Combinatorics?

Counting has been an ancient and important part of our lives. Likewise, it is also an important field when it comes to Mathematics. Now, counting objects that are less in number is an easier task but when it comes to counting large numbers of objects, it becomes a little difficult. Especially for students, the task of counting large volumes of units can be boring to say the least. So how do we speed up this process? Combinatorics is one such concept that can make the counting process easier. Mathematical calculations can be easily carried out by using this concept.

### What is Combinatorics? Combinatorial Meaning

As per the scientific definitions and concepts, combinatorial meaning can be perfectly explained as the process of counting objects in a dataset by making use of the enumeration, permutation, and combination techniques so that accurate results are obtained. Therefore, the study of combinatorics, as the name suggests, is a combination of various branches of mathematics. This is an essential application which will help students to gain a lot of knowledge from the diverse segments of mathematics. Thus, this is the overall combinatorial meaning and its concept.

The range that this concept has in its applications includes logic, statistical physics, computer science, evolutionary biology, etc. Therefore, students planning to build a career in research, development, and IT industries must have a deep rooted knowledge of Combinatorics in general.

### Combinatorics Formula

The Combinatorics Formula is a union of both the Permutation and Combination concepts. The formula for finding out the Permutation for a set of objects is as given below.

P(n,r) = $\frac{n!}{(n-r)!}$

Where n is the total number of objects

And r is the conditions considered at a given time.

A combination is an unordered arrangement of objects in a collection. The objects can be randomly arranged. They are further classified into two types; repetition and without repetition of objects. The formula for finding out the Combination for a set of objects is as given below.

C(n,r) = $\frac{n!}{(n-r)!r!}$

Where n is the total number of objects

And r is the conditions considered at a given time

The Combinatorics Formula helps to carry out mathematical operations on large datasets easily.

### What are the Combinatorics Applications?

Combinatorics deals with the counting of things in a dataset by following a particular pattern. It has its main applications when studying discrete objects.

It has its major applications in the field of Computer Science as it deals with the study and application of several programming related real-time applications that are based on the concepts of Mathematics.

### Some of the Other Combinatorics Applications are as Follows:

• Discrete Mathematics

• Discrete Harmonic Analysis

• Computer Architecture

• Scientific Research and Development

• Data Mining and Pattern Analysis

• Communications Networks and Security

• Probabilistic Methods

• Combinatorial Geometry

• Extremal Problems for Graphs and Set Systems

• Ramsey Theory

### What are Permutation and Combination?

In Mathematics, Permutation and Combination are used to describe and define a collection of individual objects. Both of them are related to the collection of objects but are different.

A permutation is an ordered arrangement of the objects in a collection. They are further classified into two types; repetition and without repetition of objects.

A combination is an unordered arrangement of objects in a collection. The objects can be randomly arranged. They are further classified into two types; repetition and without repetition of objects.

Counting is an integral part of Mathematics and is easily carried out when the number of objects to be counted is less. The process is much more difficult when large datasets have to be handled. To solve this issue and to gain accurate output, the methods of Permutation and Combination are used.

### Combinatorics Problems – Solved Example

There are various types of Combinatorics problems that students can work on. One of the most significant format of combinatorics problems has been highlighted below:

1.  Calculate the number of groups of 4 students that can be elected from the class of 28 students.

A) Given that 4 students have to be selected in a team. Since no specifications have been provided, their order should be considered as random. Therefore, the formula for Combination will be used.

C(28,4) = $\frac{28!}{(28-4)!4!}$ = 20475

There are 20475 ways in which a team of 4 students can be made out of 28 students, considering no proper order is followed.